A basis B = {e_i }_(i∈I) of an associative algebra A, over an arbitrary base field F, is called multiplicative if for any i, j ∈ I we have that e_ie_j ∈ Fe_k for some k ∈ I. The class of associative algebras admitting a multiplicative basis can be seen as a particular case of the more general class of associative algebras admitting a quasi-multiplicative basis. In the present paper we prove that if an associative algebra A admits a quasi-multiplicative basis then it decomposes as the sum of well-described ideals admitting quasi-multiplicative bases plus (maybe) a certain linear subspace. Also the minimality of A is characterized in terms of the quasi-multiplicative basis and it is shown that, under mild conditions, the above decomposition is actually the direct sum of the family of its minimal ideals admitting a quasi-multiplicative basis.
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